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# The resolution limit of a microscope is roughly equal to the wavelength of light used in producing the image . Electron microscopes use an electron beam (in place of photons) to produce much higher resolution ima ges, about 0.20 nm in modern instruments. Assuming that the resolution of an electron microscope is equal to the de Broglie wavelength of the electrons used, to what speed must the electrons be accelerated to obtain a resolution of 0.20 nm?

Solution

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Using the de Broglie equation, which is: $\lambda= \dfrac{h}{mv}$ Where:

$\lambda$= wavelength

h= Planck's constant which is 6.626$\times10^{-34}$ J $\cdot$ s

m= mass of the object

v= velocity or the speed of the object

Since the velocity is unknown, we manipulate the de Broglie equation to formulate an equation for the velocity.

$\begin{equation} v= \dfrac{h}{m\lambda} \end{equation}$

Using equation (1), we can now find the speed must the electrons be accelerated to obtain a resolution of 0.20 nm or 0.20 $\times10^{-9}$ meters. Since the mass of the electron is also constant which is 9.11$\times10^{-31}$ kilograms. Thus, $v= \dfrac{6.626\times10^{-34} \text{J $\cdot$ s}}{(9.11\times10^{-31} \text{ kg})(0.20 \times10^{-9} \text{m})}= \boxed{3636663.008 \text{ m/s or} 3.6 \times 10^6 \text{ m/s}}$

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